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Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants

Author

Listed:
  • A. Bensoussan
  • M. Crouhy
  • D. Galai

Abstract

We propose a general framework to assess the value of the financial claims issued by the firm, European equity options and warrantsin terms of the stock price. In our framework, the firm's asset is assumed to follow a standard stationary lognormal process with constant volatility. However, it is not the case for equity volatility. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. In a previous paper we studied the stochastic process for equity volatility, and proposed analytic approximations for different capital structures. In this companion paper we derive analytic approximations for the value of European equity options and warrants for a firm financed by equity, debt and warrants. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities either as a function of the stock price, or as a function of the firm's total assets. Since stock prices are observable, then for practical purposes, traders prefer to use the stock as the underlying instrument, we concentrate on valuation models in terms of the stock price. Second, we derive an exact solution for the valuation in terms of the stock price of (i) a European call option on the stock of a levered firm, i.e. a European compound call option on the total assets of the firm, (ii) an equity warrant for an all-equity firm, and (iii) an equity warrant for a firm financed by equity and debt. Unfortunately, to compute these solutions we need to specify the function of the stock price in terms of the firm's assets value. In general we are unable to specify this expression, but we propose tight bounds for the value of these options which can be easily computed as a function of the stock price. Our results provide useful extensions of the Black-Scholes model.

Suggested Citation

  • A. Bensoussan & M. Crouhy & D. Galai, 1995. "Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(1), pages 43-60.
  • Handle: RePEc:taf:apmtfi:v:2:y:1995:i:1:p:43-60
    DOI: 10.1080/13504869500000003
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    References listed on IDEAS

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    1. Galai, Dan & Masulis, Ronald W., 1976. "The option pricing model and the risk factor of stock," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 53-81.
    2. Crouhy, Michel & Galai, Dan, 1994. "The interaction between the financial and investment decisions of the firm: the case of issuing warrants in a levered firm," Journal of Banking & Finance, Elsevier, vol. 18(5), pages 861-880, October.
    3. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    4. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.),Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    5. Galai, Dan & Schneller, Meir I, 1978. "Pricing of Warrants and the Value of the Firm," Journal of Finance, American Finance Association, vol. 33(5), pages 1333-1342, December.
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    8. Alain Bensoussan & Michel Crouhy & Dan Galai, 1994. "Stochastic equity volatility related to the leverage effect," Applied Mathematical Finance, Taylor & Francis Journals, vol. 1(1), pages 63-85.
    9. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    10. Brennan, M J & Schwartz, Eduardo S, 1977. "Convertible Bonds: Valuation and Optimal Strategies for Call and Conversion," Journal of Finance, American Finance Association, vol. 32(5), pages 1699-1715, December.
    11. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    12. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    13. Geske, Robert, 1977. "The Valuation of Corporate Liabilities as Compound Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 541-552, November.
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    Citations

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    Cited by:

    1. Abdelkader Derbali, 2018. "How the default probability is defined by the CreditRisk+ model?," Working Papers hal-01696011, HAL.
    2. Jean-Guy Simonato, 2015. "New Warrant Issues Valuation with Leverage and Equity Model Errors," Journal of Financial Services Research, Springer;Western Finance Association, vol. 47(2), pages 247-272, April.
    3. Crouhy, Michel & Galai, Dan & Mark, Robert, 2000. "A comparative analysis of current credit risk models," Journal of Banking & Finance, Elsevier, vol. 24(1-2), pages 59-117, January.
    4. Abdelkader Derbali, 2018. "The credit portfolio management by structural models: A theoretical analysis," Working Papers hal-01696009, HAL.
    5. Abdelkader Derbali, 2018. "The Credit Portfolio Management by the Econometric Models: A Theoretical Analysis," Acta Universitatis Danubius. OEconomica, Danubius University of Galati, issue 14(4), pages 612-618, AUGUST.
    6. Abdelkader Derbali, 2018. "The Current Models of Credit Portfolio Management: A Comparative Theoretical Analysis," Acta Universitatis Danubius. OEconomica, Danubius University of Galati, issue 14(5), pages 184-216, OCTOBER.

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